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Let the system of equations be written in matrix form as = where is the coefficient matrix, is the vector of unknowns, and is an vector of constants. In which case, if the system is indeterminate, then the infinite solution set is the set of all vectors generated by [4]
Arguments: A: nxn numpy matrix. b: n dimensional numpy vector. omega: relaxation factor. initial_guess: An initial solution guess for the solver to start with. convergence_criteria: The maximum discrepancy acceptable to regard the current solution as fitting.
At any step in a Gauss-Seidel iteration, solve the first equation for in terms of , …,; then solve the second equation for in terms of just found and the remaining , …,; and continue to . Then, repeat iterations until convergence is achieved, or break if the divergence in the solutions start to diverge beyond a predefined level.
For example, as three parallel planes do not have a common point, the solution set of their equations is empty; the solution set of the equations of three planes intersecting at a point is single point; if three planes pass through two points, their equations have at least two common solutions; in fact the solution set is infinite and consists ...
The conjugate gradient method can be applied to an arbitrary n-by-m matrix by applying it to normal equations A T A and right-hand side vector A T b, since A T A is a symmetric positive-semidefinite matrix for any A. The result is conjugate gradient on the normal equations (CGN or CGNR). A T Ax = A T b
Indeed, multiplying each equation of the second auxiliary system by , adding with the corresponding equation of the first auxiliary system and using the representation = +, we immediately see that equations number 2 through n of the original system are satisfied; it only remains to satisfy equation number 1.
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives.
So there is a unique solution to the original system of equations. Instead of stopping once the matrix is in echelon form, one could continue until the matrix is in reduced row echelon form, as it is done in the table. The process of row reducing until the matrix is reduced is sometimes referred to as Gauss–Jordan elimination, to distinguish ...